 10.6.1: Exer. 18: Find the number.
 10.6.2: Exer. 18: Find the number.
 10.6.3: Exer. 18: Find the number.
 10.6.4: Exer. 18: Find the number.
 10.6.5: Exer. 18: Find the number.
 10.6.6: Exer. 18: Find the number.
 10.6.7: Exer. 18: Find the number.
 10.6.8: Exer. 18: Find the number.
 10.6.9: Exer. 912: Simplify the permutation.
 10.6.10: Exer. 912: Simplify the permutation.
 10.6.11: Exer. 912: Simplify the permutation.
 10.6.12: Exer. 912: Simplify the permutation.
 10.6.13: How many threedigit numbers can be formed from the digits 1, 2, 3,...
 10.6.14: Work Exercise 13 for fourdigit numbers.
 10.6.15: How many numbers can be formed from the digits 1, 2, 3, and 4 if re...
 10.6.16: Determine the number of positive integers less than 10,000 that can...
 10.6.17: If eight basketball teams are in a tournament, find the number of d...
 10.6.18: Work Exercise 17 for 12 teams.
 10.6.19: A girl has four skirts and six blouses. How many different skirtbl...
 10.6.20: Refer to Exercise 19. If the girl also has three sweaters, how many...
 10.6.21: In a certain state, automobile license plates start with one letter...
 10.6.22: Two dice are tossed, one after the other. In how many different way...
 10.6.23: A row of six seats in a classroom is to be filled by selecting indi...
 10.6.24: A student in a certain college may take mathematics at 8, 10, 11, o...
 10.6.25: In how many different ways can a test consisting of ten trueorfal...
 10.6.26: A test consists of six multiplechoice questions, and there are fiv...
 10.6.27: In how many different ways can eight people be seated in a row?
 10.6.28: In how many different ways can ten books be arranged on a shelf?
 10.6.29: With six different flags, how many different signals can be sent by...
 10.6.30: In how many different ways can five books be selected from a twelve...
 10.6.31: How many fourletter radio station call letters can be formed if th...
 10.6.32: There are 24 letters in the Greek alphabet. How many fraternities m...
 10.6.33: How many tendigit phone numbers can be formed from the digits if t...
 10.6.34: After selecting nine players for a baseball game, the manager of th...
 10.6.35: A customer remembers that 2, 4, 7, and 9 are the digits of a fourd...
 10.6.36: Work Exercise 35 if the digits are 2, 4, and 7 and one of these dig...
 10.6.37: Three married couples have purchased tickets for a play. Spouses ar...
 10.6.38: Ten horses are entered in a race. If the possibility of a tie for a...
 10.6.39: Owners of a restaurant advertise that they offer 1,114,095 differen...
 10.6.40: (a) In how many ways can a standard deck of 52 cards be shuffled? (...
 10.6.41: A palindrome is an integer, such as 45654, that reads the same back...
 10.6.42: Each of the six squares shown in the figure is to be filled with an...
 10.6.43: The graph of has a horizontal asymptote of . Use this fact to find ...
 10.6.44: (a) What happens if a calculator is used to find ? Explain. (b) App...
Solutions for Chapter 10.6: Permutations
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 10.6: Permutations
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Since 44 problems in chapter 10.6: Permutations have been answered, more than 34828 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Chapter 10.6: Permutations includes 44 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.